minus of ff tilde alpha left-parenthesis right-parenthesis xi plus of ff tilde alpha left-parenthesis right-parenthesis xi times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis plus plus Ì‚ xi of alpha alpha left-parenthesis right-parenthesis xi of ff left-parenthesis right-parenthesis xi 2nd Row 1st Column Blank 2nd Column equals vertical-bar vertical-bar vertical-bar vertical-bar integral integral xi plus plus xi rho minus minus times times times d of alpha alpha left-parenthesis right-parenthesis t of ff left-parenthesis right-parenthesis t times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis plus plus Ì‚ xi of alpha alpha left-parenthesis right-parenthesis xi of ff left-parenthesis right-parenthesis xi 3rd Row 1st Column Blank 2nd Column less-than-or-slanted-equals vertical-bar vertical-bar vertical-bar vertical-bar integral integral xi plus plus xi rho minus minus times times times d of alpha alpha left-parenthesis right-parenthesis t of ff left-parenthesis right-parenthesis t times times left-bracket right-bracket minus minus times times alpha left-parenthesis right-parenthesis plus plus xi rho of alpha alpha left-parenthesis right-parenthesis xi of ff left-parenthesis right-parenthesis xi 4th Row 1st Column Blank 2nd Column plus vertical-bar vertical-bar vertical-bar vertical-bar minus minus times times left-bracket right-bracket minus minus times times alpha left-parenthesis right-parenthesis plus plus xi rho of alpha alpha left-parenthesis right-parenthesis xi of ff left-parenthesis right-parenthesis xi times times left-bracket right-bracket minus minus of alpha alpha left-parenthesis right-parenthesis plus plus Ì‚ xi of alpha alpha left-parenthesis right-parenthesis xi of ff left-parenthesis right-parenthesis xi less-than epsilon EndLayout"/>
and the proof is complete.
With Theorem 1.49 at hand, the next corollary follows immediately.
Corollary 1.50: The following statements hold:
1 If and , then .
2 If and , then .
Next, we state a uniform convergence theorem for Perron–Stieltjes vector integrals. A proof of such result can be found in [210, Theorem 11].
Theorem 1.51: Let and , , be such that the Perron–Stieltjes integral exists for every and uniformly in . Then, exists and
We finish this subsection by presenting a Grönwall‐type inequality for Perron–Stieltjes integrals. For a proof of it, we refer to [209, Corollary 1.43].
Theorem 1.52 (Grönwall Inequality): Let be a nondecreasing left‐continuous function, and . Assume that is bounded and satisfies
Then,
Other properties of Perron–Stieltjes integrals can be found in Chapter 2, where they appear within the consequences of the main results presented there.
1.3.3 Integration by Parts and Substitution Formulas
The first result of this section is an Integration by Parts Formula for Riemann–Stieltjes integrals. It is a particular consequence of Proposition 1.70 presented in the end of this section. A proof of it can be found in [126, Theorem II.1.1].
Theorem 1.53 (Integration by Parts): Let be a BT. Suppose
1 either and ;
2 or and .
Then, and , that is, the Riemann–Stieltjes integrals and exist, and moreover,
Next, we state a result which is not difficult to prove using the definitions involved in the statement. See [72, Theorem 5]. Recall that the indefinite integral of a function
Theorem 1.54: Suppose and is bounded. Then and
(1.3)
If, in addition, , then .
By Theorem 1.47, the Perron–Stieltjes integral
